Page 245 - Compact Numerical Methods For Computers

232 Compact numerical methods for computers
The second residual is the likely form in which the demand function would be
estimated. To obtain a concrete and familiar form, substitute
q =b 1 p = b 2 K =1
= 1·5 b = 1·2 Z = exp(2)
so that

f =ln(b )-2+1·2ln( b ).
1
2
2
Now minimising the sum of squares

should give the desired solution.
The Marquardt algorithm 23 with numerical approximation of the Jacobian as
in §18.2 gives
p =b =2·09647 q=b =3·03773
1
2
with S=5·28328E-6 after five evaluations of the Jacobian and 11 evaluations of
S. This is effectively 26 function evaluations. The conjugate gradients algorithm
22 with numerical approximation of the gradient gave
p=2·09739 q=3·03747 S=2·33526E-8
after 67 sum-of-squares evaluations. For both these runs, the starting point
chosen was b =b =1. All the calculations were run with a Data General NOVA
2
1
in 23-bit binary arithmetic.
Example 18.3. Magnetic roots
Brown and Gearhart (1971) raise the possibility that certain nonlinear-equation
systems have ‘magnetic roots’ to which algorithms will converge even if starting
points are given close to other roots. One such system they call the cubic-
parabola:

To solve this by means of algorithms 19, 21, 22 and 23, the residuals

were formed and the following function minimised:

The roots of the system are
R : b =0= b 2
1
1
R : b =0= b
2 1 2
R : b =-0·75 b =0·5625.
3
1
2